On the finite dimensional approximation of the Kuratowski-embedding for compact manifolds

TL;DR

This paper proves the existence of finite-dimensional approximations of the Kuratowski-embedding for compact manifolds, providing bounds based on curvature and extending Gromov's work on systolic inequalities.

Contribution

It offers a detailed proof of finite-dimensional approximations of the Kuratowski-embedding, including quantitative bounds related to manifold curvature.

Findings

Finite-dimensional approximations exist for Kuratowski-embedding.

Bounds on the dimension depend on curvature properties.

The approach follows and extends ideas suggested by Larry Guth.

Abstract

In the proof of his systolic inequality, Gromov uses an isometric embedding of a Riemannian manifold M into the Banach space of bounded functions on M, the so-called Kuratowski-embedding. Subsequently, it was shown by different authors that the Kuratowski embedding can be approximated by bi-Lipschitz embeddings into finite-dimensional Banach spaces. We give a detailed proof for the existence of such finite-dimensional approximations along the lines suggested by Larry Guth and go on to discuss quantitative aspects of the problem, establishing for the dimension of the Banach space a bound which depends on curvature properties of the manifold.